On Saturday, November 17, 2012 6:10:00 PM UTC+8, Daniel J. Greenhoe wrote:> Closure in topological space is defined using at least two different ways in the literature:> > 1. cl(A) is the intersection of all closed sets containing A.> > 2. cl(A) is the intersection of all neighborhoods containing A, where a neighborhood is any set containing an open set (an element of the topology).> > > > Examples of authors who use 1 include Kelley, Munkres, Thron, and McCarty.> > Examples of authors who use 2 include Mendelson and Aliprantis & Burkinshaw.> > > > My question is, one definition considered to be more "standard" than the other (from my very limited survey, 1 might seem more standard). > > > > Aliprantis/Burkinshaw hints that 2 is influenced by metric space theory.> > > > I might guess that there are other definitions possible (hence the "Kuratowski closure axioms"?)> > > > Pointers to good references are especially appreciated.> > > > Many thanks in advance,> > Dan